Problem 1 - Algebra for College Students

Question

For Problems \(1-22\), graph each of the polynomial functions. $$ f(x)=-(x-3)^{3} $$

Step-by-Step Solution

Answer

Graph the reflected cubic function centered at \((3, 0)\).

Step 1: Identify the Base Function

The base function is \((x-3)^3\). This is a cubic function centered at \(x = 3\). Without any transformations, it would pass through the point \( (3,0) \) and have the typical cubic shape, increasing rapidly on either side of its center.

Step 2: Understand the Transformation

The function is given as \(f(x) = -(x-3)^3\). The negative sign in front of the cubic expression indicates a vertical reflection across the x-axis. This means that the basic cubic shape will be flipped upside down.

Step 3: Identify Key Points of the Function

To graph the function, identify key points such as the vertex and other critical points. For \(f(x) = -(x-3)^3\), the vertex (or turning point) is at \((3, 0)\). Consider additional points to better understand the curve, such as \(f(2) = -(-1)^3 = 1\) and \(f(4)=-(1)^3 = -1\).

Step 4: Sketch the Graph

Plot the key points identified, such as \((3, 0)\), \((2, 1)\), and \((4, -1)\). Draw a smooth upside-down cubic curve that passes through these points. The graph should asymptotically approach the general cubic shape, starting upwards on the left and curving downwards sharply through \((3,0)\) before heading downwards on the right.

Key Concepts

Cubic Functions

Cubic functions are a special type of polynomial function where the highest degree of any term is three. This gives these functions their name and unique characteristics. From a general perspective, a standard cubic function has the form \[ f(x) = ax^3 + bx^2 + cx + d \]where \(a\), \(b\), \(c\), and \(d\) are constants, and \(a eq 0\).
The graph of a basic cubic function, such as \(f(x) = x^3\), forms an 'S' shape curve when plotted on a coordinate plane. This curve has the following properties:

• It passes through the origin, unless shifted.
• It is symmetric about its center point.
• As \(x\) goes to positive or negative infinity, the function's value increases or decreases without bound.

When graphing a cubic function, noticing these characteristics can be very helpful. More complicated cubic functions that include transformations exhibit varied behavior based on these basic principles.

Transformations of Functions

Transformations involve changing the position or the shape of the graph without altering its functional form. When discussing functions, these changes might be translations, reflections, stretches, or compressions. For cubic functions, transformations are pivotal in accurately sketching their graphs.The function \(f(x) = (x - h)^3\) demonstrates a horizontal shift to the right by \(h\) units.
This adjustment moves the typical center of the cubic equation from the origin to another specific point on the \(x\)-axis, such as at \(x = 3\) for \((x - 3)^3\). Common transformations include:

• Translation: Moves the graph up, down, left, or right.
• Reflection: Flips the graph over a line, such as the x-axis.
• Scaling: Stretches or compresses the graph vertically or horizontally.

By understanding transformations, graphing complicated polynomial functions becomes a matter of altering a familiar base function to its new configuration.

Vertical Reflections

Reflecting a graph vertically means flipping it over the x-axis. This is simply done by multiplying the function by -1. For example, changing \((x - 3)^3\) to \(-(x - 3)^3\) reflects the cubic function. The shape of the cubic curve becomes inverted.
In practical terms, before the reflection, the function starts rising sharply from left to right, decreases through the origin, and then continues rising. After a vertical reflection:

• The line going up now goes downward.
• The curve descends sharply through its center (or vertex).
• What was previously increasing at the end now decreases.

The application of a vertical reflection effectively rotates the cubic function around the x-axis. This transformation doesn't affect the x-intercepts or location of the center, just the orientation of the graph. Vertical reflections are integral when transforming and graphing polynomial functions confidently.

Key Points of Functions

Identifying key points on a graph is vital for understanding a function's behavior, especially for complex equations like cubic functions. These points give insight into the graph's shape and help with accurate plotting. Key points typically include the vertex or central point and additional points that illustrate the curve effectively.For the function \(f(x) = -(x-3)^3\), the key point is located at \( (3, 0) \), which acts as its turning point. Finding other points such as \(f(2)\) and \(f(4)\) allows us to determine the direction and thickness of the curve:

• \(f(3) = 0\) since replacing \(x\) with 3 yields zero.
• \(f(2) = 1\) results from \(-(x-3)^3 = -(-1)^3 = 1\).
• \(f(4) = -1\) provides \(-(x-3)^3 = -(1)^3 = -1 \).

These calculations guide us in accurately drawing the graph and help visualize how the function behaves around the vertex and near intercepts. Key points are crucial for mastering graphing and comprehending polynomial equations fully.